A compact subset K of Euclidean space R^n is called metrically removable if any two points a,b of its complement can be joined by a curve that is disjoint from K and has length arbitrarily close to |a-b|. Every set of zero (n-1)-dimensional measure is metrically removable, but not conversely. Metrically removable sets can even have positive n-dimensional measure.
I will describe some properties of metrically removable sets and outline a proof of the following fact: totally disconnected sets of finite (n-1)-dimensional measure are metrically removable. This answers a question raised by Hakobyan and Herron in 2008.
Joint work with Sergei Kalmykov and Tapio Rajala.

FridayJanuary 26, 201811:00 AM - 12:00 AM P-131

Frank Thorne, South CarolinaLevels of Distribution for Prehomogeneous Vector Spaces

This will be a continuation of Thursday's talk, where I will explain multiple approaches to the lattice point counting problem. The quantitatively strongest estimates all involve Fourier analysis in some guise, which turns out to have nice interplay with the action of the group.

We prove a uniqueness of blowups result for isolated singular points in the free boundary of minimizers to the Alt-Caffarelli functional. The key tool is a (log-)epiperimetric inequality, which we prove by means of two Fourier expansions (one on the function and one on its free boundary).

If time allows we will also explain how this approach can be adapted to (re)prove old and new regularity results for area-minimizing hypersurfaces.